Question

determining congruence
the vertices a(1, - 2), b(1, - 4), and c(5, - 2) form a triangle. the vertices a(- 3, 1), b(- 1, 1) and c(- 3, 5)
are the image of the triangle after a sequence of transformations. which sequence of transformations could produce the
image from the pre - image?
a 90° clockwise
rotation about the
origin and then a
reflection over the
y - axis
a reflection over
the x - axis and
then a reflection
over the y - axis
a reflection over
the y - axis and then
a 90° clockwise
rotation about the
origin
a 90°
counterclockwise
rotation about the
origin and then a
translation left 5
units

Explanation:

Step1: Find the coordinate transformation rules

• For a \(90^{\circ}\) clockwise rotation about the origin \((x,y)\to(y, - x)\).
• For a reflection over the \(y -\)axis \((x,y)\to(-x,y)\).
• For a \(90^{\circ}\) counter - clockwise rotation about the origin \((x,y)\to(-y,x)\).
• For a translation left \(a\) units \((x,y)\to(x - a,y)\).

Step2: Apply the transformations in the first option

First, consider the first option: a reflection over the \(x -\)axis \((x,y)\to(x,-y)\), then a reflection over the \(y -\)axis \((x,-y)\to(-x,-y)\).
For point \(A(1,-2)\): After reflection over \(x -\)axis \((1,2)\), then reflection over \(y -\)axis \((-1,2)
eq A'(-3,1)\).

Step3: Apply the transformations in the second option

Second option: \(90^{\circ}\) clockwise rotation about the origin \((x,y)\to(y,-x)\), then reflection over the \(y -\)axis \((y,-x)\to(-y,-x)\).
For \(A(1,-2)\): After \(90^{\circ}\) clockwise rotation \((-2,-1)\), then reflection over \(y -\)axis \((2,-1)
eq A'(-3,1)\).

Step4: Apply the transformations in the third option

Third option: reflection over the \(y -\)axis \((x,y)\to(-x,y)\), then \(90^{\circ}\) clockwise rotation about the origin \((-x,y)\to(y,x)\).
For \(A(1,-2)\): After reflection over \(y -\)axis \((-1,-2)\), then \(90^{\circ}\) clockwise rotation \((-2,-1)
eq A'(-3,1)\).

Step5: Apply the transformations in the fourth option

Fourth option: \(90^{\circ}\) counter - clockwise rotation about the origin \((x,y)\to(-y,x)\), then translation left \(5\) units \((-y,x)\to(-y - 5,x)\).
For \(A(1,-2)\): After \(90^{\circ}\) counter - clockwise rotation \((2,1)\), then translation left \(5\) units \((2-5,1)=(-3,1)\).
For \(B(1,-4)\): After \(90^{\circ}\) counter - clockwise rotation \((4,1)\), then translation left \(5\) units \((4 - 5,1)=(-1,1)\).
For \(C(5,-2)\): After \(90^{\circ}\) counter - clockwise rotation \((2,5)\), then translation left \(5\) units \((2-5,5)=(-3,5)\).

Answer:

The fourth option (a \(90^{\circ}\) counterclockwise rotation about the origin and then a translation left \(5\) units)